## Abstract

We derive the closed system of averaged magnetohydrodynamic (MHD) equations for general oscillating flows. The used small parameter of our asymptotic theory is the dimensionless inverse frequency, and the leading term for a velocity field is chosen to be purely oscillating. The employed mathematical approach combines the two timing method and the notion of a distinguished limit. The properties of commutators are used to simplify calculations. The derived averaged equations are similar to the original MHD equations, but surprisingly (instead of the commonly expected Reynolds stresses) a drift velocity plays a part of an additional advection velocity. In the special case of a vanishing magnetic field $h\equiv 0$, the averaged equations produce the Craik–Leibovich equations for Langmuir circulations (which can be called ‘vortex dynamo’). We suggest that, since the mathematical structure of the full averaged equations for $h\neq 0$ is similar to those for $h =0$, these full equations could lead to a possible mechanism of MHD dynamo, such as the generation of the magnetic field of

the Earth.

the Earth.

Original language | English |
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Pages (from-to) | 51-61 |

Number of pages | 11 |

Journal | Journal of Fluid Mechanics |

Volume | 698 |

Issue number | n/a |

Early online date | 1 Apr 2012 |

DOIs | |

Publication status | Published - May 2012 |

## Keywords

- dynamo theory, general fluid mechanics, magnetohydrodynamics